Vector-valued function

Summary

A vector-valued function is a function whose values are vectors, for example $f \colon D \to \mathbb{R}^m$ with $D \subseteq \mathbb{R}^n$ and $m \ge 2$. Such a function has $m$ component functions $f_1, \dots, f_m \colon D \to \mathbb{R}$, with $f(x) = (f_1(x), \dots, f_m(x))$ for each $x \in D$. An example is $r(t) = (\cos t, \sin t, t)$ for $t \in \mathbb{R}$, whose values trace a helix in the Euclidean space $\mathbb{R}^3$.

Remarks

Limits, continuity and derivatives of a vector-valued function into $\mathbb{R}^m$ can be computed one component at a time. When the domain is an interval of $\mathbb{R}$, the function is a parametrised curve.

Explanations

L1 human

A vector-valued function gives an arrow, or a point, for each input, not a single number. If $t$ is time, then $r(t)$ can be the position of a moving particle in space.